The billionaire George Soros and other liberal donors will bankroll a new $15 million campaign to mobilize Latinos and other immigrants this fall, hoping to channel outrage at the political rhetoric of Donald J. Trump and other Republicans into a surge of votes for Democratic candidates in November.

Strategists involved said the new spending would be the largest Democratic voter-turnout effort ever devoted exclusively to Latino and immigrant voters. Most of the money will be spent through organizations in Colorado, Florida and Nevada, states with large or growing Latino and Asian populations that will be pivotal in the presidential race and in the battle for control of the Senate.

The outreach, which will be coordinated through a new “super PAC” called Immigrant Voters Win PAC, will be more explicitly political and partisan than past efforts, the strategists said: The goal was to not only turn out committed Latinos already voting Democratic but also find and persuade immigrant swing voters. Ultimately, organizers hope to get at least 400,000 new Democratic voters to the polls in November.

“This is really taking the gloves off,” said Cristóbal Alex, the president of the Latino Victory Project, one of several national pro-immigration or Hispanic-oriented groups working with the super PAC. “From the first day he attacked us, he called us rapists and thieves,” Mr. Alex said of Mr. Trump. “We could have a giant wall built and millions of families broken apart. The country is on the precipice.”

The effort comes amid signs of a Democratic enthusiasm gap that has worried some of the party’s leading strategists. While Republican voters are turning out in droves, often to vote for Mr. Trump, millions of Democratic voters have sat out the party’s primary between Hillary Clinton and Senator Bernie Sanders of Vermont.

Conservatives, to be sure, are hardly sitting by idly as Soros and Co. begin their Latino outreach.

Libre Initiative, which is in large part funded by the Koch brothers, reportedly is spending some $10 million to mobilize Latino turnout. The organization’s approach is to have a grassroots presence in states where Latinos live in large numbers or where there is a fast-growing community.

The group promotes conservative principles while providing help to people obtaining driver's licenses or preparing for GED exams, among other things.

Trump wasted no time in laying out his hard-line immigration proposals. When he announced he was running for president, he condemned Mexico for dumping its worst citizens, as he put it, on the U.S. He vowed to build a wall along the border and to have a kind of deportation army that would track down and remove the estimated 11 million undocumented people living in the United States.

Soros told the Times in an email that he had been rattled by what he viewed as the xenophobic tone of the Republicans running for president. He was upset by the call by Trump and others to ban Muslim refugees from the United States.

Soros has a long history of contributing millions to liberal political causes, and pockets don't get much deeper than his. He ranked No. 23 in the latest Forbes richest men list.

He told the Times, “The intense anti-immigrant and anti-Muslim rhetoric that has been fueled by the Republican primary is ... harmful to our democracy and to our national interests,” Soros said. “There should be consequences for the outrageous statements and proposals that we’ve regularly heard.”

The Times said participants in the new campaign are responding to criticism by Latinos about the lack of financial support from wealthy liberals.

“One reason we had lower turnout is because of historical underinvestment in our community,” Alex said to the Times. “Folks who want a progressive vision of the country have to match what is happening on the right. Now we are seeing a recognition by certain donors of the importance of our vote.”